Parametric Methodology in Optimization

当优化问题所依赖的参数受到扰动时，优化问题的解会发生什么样的变化，这一问题不仅在评估数学模型的稳定性方面很重要，而且在计算程序的设计和论证方面也很重要。 参数依赖性问题也是在动态优化中有效利用“剩余成本”函数的关键，无论是在最优控制还是在随机规划中。 该项目将推进处理这种依赖关系的方法，并探讨其对计算的影响。将应用变分分析中的最新工具，这是必不可少的，因为参数依赖关系中固有的非光滑性。将在有限维和无限维的广泛框架内研究最优性条件的替代形式，特别是扰动鲁棒性。 凸动力优化中的成本函数将从确定性连续时间模型（与Hamilton-Jacobi理论的联系）和随机离散时间模型（其中数值势将占主导地位）的新角度进行研究。

Questions about what happens to a solution to a problem of optimization when the parameters on which the problem depends are perturbed are important not only in assessing the stability of mathematical models, but also in the design and justification of computational procedures. Issues of parametric dependence are also the key to effective utilization of `cost-to-go' functions in dynamical optimization, whether in optimal control or stochastic programming. This project would advance the methodology for dealing with such dependence and explore its consequences for computation.The latest tools in variational analysis, which are essential becauseof inherent nonsmoothness in parametric dependence, would be applied.Alternative forms of optimality conditions, focused especially on perturbational robustness, would be investigated in a broad framework,both finite- and infinite-dimensional. Cost-to-go functions in convexdynamical optimization would be studied from the new perspective both in deterministic continuous-time models, with their connections to Hamilton-Jacobi theory, and in stochastic discrete-time models, wherethe numerical potential will dominate.